preprint - Gipsa-lab

A Parameter Varying Observer for the
Enclosed Mass in a Spark Ignited Engine
Maria Adelina Rivas Caicedo, ∗ Olivier Sename ∗
Emmanuel Witrant ∗ Christian Caillol ∗∗ Pascal Higelin ∗∗
∗
Université Joseph Fourier, INP /CNRS GIPSA-lab. 11 rue de
mathematiques, BP 46, 38402 Saint Martin d’Hères Cedex, France,
∗∗
(maria-adelina.rivas-caicedo, emmanuel.witrant,
olivier.sename)@gipsa-lab.grenoble-inp.fr
PRISME, Polytech’Orléans, Site Vinci JOULE 112, 8 Rue Leonardo
Da Vinci, 45072 Orléans Cedex, France, (pascal.higelin,
christian.caillol)@univ-orleans.fr
Abstract: In this paper, a high gain observer is designed for a parameter varying polytopic
model to estimate the enclosed mass in the combustion chamber of a spark ignited engine.
The high gain strategy allows the design of an observer that handles the uncertain part of the
system. The observer uses the cylinder pressure measurement during the compression stroke to
estimate the enclosed mass. An engine compression model is used as a virtual engine to build
the observer. The results of the observer are compared to the virtual engine model and a good
agreement between the observed variables and the model was obtained.
1. INTRODUCTION
The enclosed mass estimation problem is an interesting
and challenging task in the engine control. Indeed, an
accurate estimation of the enclosed mass would permit a
better control of the associated fuel injection and a better
treatment of the pollutant residuals [Butt and Bhatti,
2008].
In automotive control, the variables in the air path are
typically used to compute the cylinder characteristics,
such as the in-cylinder load and residual mass fraction.
Such strategies are usually designed with static approximations (e.g., see [Muller, 2008], [Kang and Chang, 2009],
[Senecal et al., 1996] and [Fox et al., 1993]). Alternatively,
closed loop observer schemes have also been developed
to estimate the engine load during the admission stroke.
In [Stotsky and Eriksson, 2002], the uncertainties of the
measurements in the intake manifold are introduced and
an adaptive learning algorithm is used to track the error
between the measured pressure and the estimation. In
[Kerkeni et al., 2010], a periodic observer for a class of
non-linear models in the discrete Takagi-Sugeno form is
designed, using the variables on the engine intake manifold. Both works propose suitable methods to estimate the
enclosed mass in the combustion chamber, by computing
the in-cylinder mass in the intake manifold, before the
valves closure. Other interesting approach can be found in
[Chauvin et al., 2008], where the estimation and to control
of the masses entering in the cylinders for a diesel Homogeneous Charge Compression Ignition (HCCI) engine. The
masses are directly related to the intake manifold pressure,
compositions and flow-rates. A nonlinear observer for periodic systems is used as estimation strategy.
The objective of this work is to estimate the total enclosed
mass in the combustion chamber, which corresponds to the
total air load plus the residual mass after the inlet valve
closure (IVC). In a previous work [Rivas et al., 2012], the
authors have proposed a nonlinear high gain observer to
estimate the enclosed mass during the compression and
combustion strokes using the cylinder pressure measurement. Compared to the previous work, in this paper, the
high gain observation strategy is applied to a parameter
varying system, to estimate the cylinder temperature during the compression stroke. The enclosed mass is computed
using the observed temperature and the cylinder pressure
measurement.
The challenge on using a parameter varying polytopic
model is the fact that the varying parameters depend
on the estimated state and nonlinear terms are added to
the system, thus the estimation error contains uncertain
elements. Solutions to this class of problems have been explored in [Daafouz et al., 2008], [Maurice et al., 2010] and
[Bara et al., 2010]. In those works, interesting strategies
to relax the mismatch due to the parameters uncertainties
are proposed. The global asymptotical stability of the
observer is assured, but for the application considered in
this work, this criterion is not enough as the goal is to
fully eliminate the mismatch between the observed state
and the system states. The contribution of this work is to
include a high gain strategy in a linear parameter varying
(LPV) observer.
Including the high gain technique improves the observer
performance, vanishing the effect of the uncertainties due
to the estimated states. It is important to notice that the
design of this high gain LPV observer is an open problem.
A related work can be found in Gérard et al. [2010],
where a high gain observer design is used to consider
the level of disturbance attenuation of an LPV functional
filter for bilinear systems with a disturbance attenuation
specification. The observer gain is written as a function of
the estimated state and the high gain observer is computed
using Linear Matrix Inequalities (LMI) techniques. In this
work, the high gain approach is used to ensure the stability
of the filtering error and to optimize the disturbance
attenuation.
This paper is structured as follows. In Section 2, the physical engine model for the compression stroke is designed
to build the observer. This model is transformed into an
equivalent parameter varying system in Section 3. In Section 4 the high gain observer is implemented: the cylinder
enclosed mass is computed in this stage. Simulations of
the observed variables are compared to a validated virtual
engine to support the results of the observer.
dp(t) = −
r
+1
cv
and replacing m(t) =
dynamics is:
dT (t) = −
r
dV (t)
p(t) −
Qth (t)
V (t)
V (t)cv
p(t)V (t)
rT (t)
(6)
in (3), the temperature
rdV (t)
r
T (t)
T (t) −
Qth (t)
cv V (t)
cv V (t)
p(t)
(7)
The wall losses Qth (t) are modeled using the reduced
model provided in [Rivas et al., 2012]:
2. COMPRESSION MODEL
Qth (t) = Aw (t)ωp(t)(k1 T (t) − k0 Tw )
The engine compression parameter varying model is based
on the engine model proposed by [Rivas et al., 2012]. The
combustion chamber is considered as a unique open system
and a uniform in-cylinder pressure is assumed. Only the
compression stroke, after the inlet valve closure (IVC)
is considered. During the engine compression cycle, no
energy is transferred between the cylinder and the inlet
and outlet ports. The energy equation for the cylinder is
inferred from the first thermodynamical principle:
where Aw (t) is the wall transfer area, ω is the engine speed,
k1 and k0 are tuning constants and Tw is the cylinder wall
temperature.
dU (t) = −Qth (t) − p(t)dV (t)
(1)
U (t) is the internal energy of the cylinder gas mixture,
Qth (t) expresses the heat transfer of the cylinder contents
to the surroundings and p(t) is the cylinder pressure,
dV (t) is the variation of the cylinder volume and p(t)dV (t)
corresponds to the work delivered by the piston.
(8)
Defining p(t) = x1 and T (t) = x2 in equations (6) and (7),
and replacing the heat wall losses by (8), the state space
system is written as:
ẋ1 = −
r
dV (t)
+1
x1
cv
V (t)
r
Aw (t)ωx1 (k1 x2 − k0 Tw )
(9)
cv V (t)
rdV (t)
r
ẋ2 = −
x2 −
x2 Aw (t)ω(k1 x2 − k0 Tw )
cv V (t)
cv V (t)
−
Assuming that the specific heat parameter cv is constant,
the left hand side of (1) can be written as:
dU (t) = m(t)cv dT (t)
(2)
where m(t) is the total mass of all the species in the
cylinder and T (t) corresponds to the temperature of the
gas. As the valves are closed, m(t) remains constant and
it is equivalent to the enclosed mass.
Solving (1) and (2) for dT (t), an ordinary differential
equation governs the system temperature dynamics during
the compression:
dT (t) =
1
(−p(t)dV (t) − Qth (t))
m(t)cv
Fig. 1. Cylinder pressure. BMEP=18 bar, N=1500 rpm.
(3)
where V (t) is the gas volume. The ideal gas law is used to
find the dynamics of p(t):
p(t)V (t) = rmT (t)
(4)
where r, is the specific gas constant. Taking the derivative
of (4) leads to:
dp(t) =
rmdT (t) rT (t)mdV (t)
−
V (t)
V (t)2
(5)
Using equations (3) and (5), the in-cylinder pressure
dynamics during the compression is modeled as:
The compression model is tested taking as reference the
measurements of a 1.2 liters engine. The data to fit the
model is the cylinder pressure. The results presented
in this paper correspond to a test performed at N =
1500 rpm and BMEP = 18 bar.
A good agreement between the measurements and the
compression model is obtained, thus the model can be
used as virtual engine to build the observer. Results of
the validated model are shown in Figure 1. Further results
are shown in Figure 2.
3. PARAMETER VARYING POLYTOPIC SYSTEM
REPRESENTATION
The model (9) can be written in the following LPV model:
considered for A(ρ(x)), since the bi-linear term x1 x2 is
present. However, x1 must be included in the parameters
in order to be consistent with the triangular additive form
required for the high gain observer design.
4. HIGH GAIN OBSERVER
The idea is to extend the notion of a high gain observer
to LP V systems under a polytopic representation. For a
nonlinear system described in additive triangular form as:
ẋ = A0 x + ψ(x, u)
(15)
y = C0 x
where
"
Fig. 2. Cylinder pressure. BMEP=2 bar, N=1200 rpm.
ẋ = A(ρ(x))x + φ(x),
y = Cx
A0 =
(10)
0 a21
0
0 ... an−1×n
0 0
0
"
A(ρ(x)) =
#
r
ρ(x)
,
cv
0
0
0
ρ(x) = −
C
= [1 0]
1
ωAw (t)k1 x1
V (t)
(11)
(12)
and
φ(x) =
(13)

r
dV (t)
r
 − cv + 1 V (t) + cv V (t) ωAw (t)k0 Tw x1 
 
 r

1
dV (t)
−
ωAw (t)(k1 x2 − k0 Tw ) x2
−
cv
V (t)
V (t)

In this study, ρ(x) is assumed to be bounded in the convex
set [ρ, ρ] and ρ 6= 0, it allows to represent the system
(10) in a polytopic approach. Due to the fact that (11)
is a quasi-LPV system since ρ(x) depends on the state x,
this assumption induces that the system state is belonging
to a bounded set Γ ⊂ Rn . Finally φ(x) is a Lipschitz
continuous function. Under those conditions, the matrix
A(ρ(x)) can be written in the form
A(ρ(x)) =
M
X
a nonlinear high gain observer can be obtained using the
following theorem:
Theorem 1. ([Besancon, 2007]). If ψ(x, u) is globally Lipschitz in x and u, thus kψ(x, u) − ψ(x̂, u)k < δkx − x̂k and
such that ∂ψi (x, u)/∂xj = 0, for j ≥ i + 1, 1 ≤ i, j ≤ n,
then System (15) admits an observer of the form:
"
#
λ 0 0
˙x̂ = A0 x̂ + ψ(x̂, u) + 0 ...
L0 (C0 x̂ − y) (17)
0 0 λn
with L0 such that A0 − L0 C0 is stable and λ, ..λn large
enough.
The idea of this observer is to use the uniform observability
to weight a gain based on the linear part, so as to make
the linear dynamics of the observer error to dominate the
nonlinear one ([Ljung, 1999]).
4.1 High Gain Observer for LPV systems
From this part of the document, the temporal dependence
notation of the variables is omitted and ρ(x) = ρx . Given
the system (10), an observer for the state x is proposed as:
x̂˙ =A0 (ρ̂x )x̂ + ΛL0 (ρ̂x )(y − ŷ) + φ(x̂)
ŷ =C0 x̂
αi (ρ(x))Ai
(16)
[ 1 0 0... ]
C0 =
where
#
(14)
i=1
A0 (ρ̂x ) =
M = 2, is the number of vertexes of the polytope formed
by the extremes of the varying parameter ρ(x). αi (ρ(x)) ∈
PN
R is a scheduling function such that i=1 αi (ρ(x)) = 1
and the matrices Ai ∈ Rn×n .
N
X
(18)
αi (ρ̂x )Ai
(19)
αi (ρ̂x )Li
(20)
i=1
L0 (ρ̂x ) =
N
X
i=1
The goal is to observe the state x2 through the measurement x1 . It is assumed that the uncertainty in x is
bounded, thus |ρ(x) − ρ(x̂)| < ∆.
Remark 1. It is important to recall that (11) is written in
a state space form that allows the design of a high gain
observer. Such form is referred to as additive triangular
and is detailed next. Two symmetrical choices can be
and

λ 0 0
 0 λ2 0

Λ=

....
n
0 0 λ

where λ ∈ R and λ > 1.
(21)
The dynamics of the estimation error e = x− x̂ is governed
by:
following inequality must be verified to ensure the error
(t) converges to 0:
h
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P +
i
P A0 (ρ̂x ) − L0 (ρ̂x )C0 (t) + 2δ(t)T P (t)
ė = (A0 (ρ̂x ) − ΛL0 (ρ̂x )C0 )e + φ(x) − φ(x̂) + vx (22)
λ(t)T
where vx = (A0 (ρx ) − A0 (ρ̂x ))x, is considered as a bilinear
perturbation L2 bounded.
+vx (t)T P Λ−1 (t) + (t)T Λ−1 P vx (t) < 0
Proposition 1. Consider the quasi-LPV system (10) and
assume hat φ(x(t)) is a Lipschitz continuous function such
that ||φ(xa (t)) − φ(xb (t))|| < δ||xa (t) − xb (t)||, δ >
0, xa , xb ∈ Dx . If there exists the state feedback L0 (ρ̂x ),
the matrices P = P T such that P > 0 and Λ defined
in Equation (21), such that the following inequality is
satisfied for all ρx ∈ [ρ, ρ]:
h



iT
h
P + P A0 (ρ̂x )
A0 (ρ̂x ) − ΛL0 (ρ̂x )C0
i
−ΛL0 (ρ̂x )C0 + 2δP + I
P
P
Considering that vx (t) is L2 bounded, the application of
the Bounded Real Lemma leads to the inequality:
G
Λ−1 P
<0
(29)
Λ−1 P −γ 2 I
where


<0

−γ 2 I
G=
(30)
h
T
i
λ A0 (ρ̂x ) − L0 (ρ̂x )C0 P + P A0 (ρ̂x ) − L0 (ρ̂x )C0
+2δP + I
(23)
for some constant γ > 0, then (18) is an observer for
System (10) and the estimation error is asymptotically
stable and satisfies ||e(t)||2 < Λγ||vx (t)||2 .
Proof : To prove the stability, the auxiliary variable z(t) =
Λ−1 x(t) is introduced and the error dynamics (t) = z(t)−
ẑ(t) is considered:
h
i
(t)
˙ = Λ−1 A0 (ρ̂x ) − ΛL0 (ρ̂x )C0 Λ +
(24)
Λ−1 φ(x(t)) − φ(x̂(t)) + Λ−1 vx (t)
Choosing the Lyapunov function V (t) = (t)T P (t) where
P = P T > 0, the following inequality must be satisfied in
order to ensure the observer stability:
therefore it is verified that ||(t)||2 < γ||vx (t)||2 for all
ρx ∈ [ρ, ρ]. Since (t) = Λ−1 e(t), (30) becomes ||e(t)||2 <
Λγ||vx (t)||2 and Proposition 1 is proved.
5. DESIGN PROCEDURE
According to Proposition 1, for the observer in (18),
Inequality (23) must be accomplish to guarantee the
observer stability. However, as there is a nonlinear relation
between the parameter Λ and the gain L0 (ρ̂x ), it is not
possible to use an LMI solver to obtain the observer
feedback. Nevertheless, it has also been proved that if (28)
is accomplished, Proposition 1 is satisfied. Thus, using this
result, the procedure to obtain Λ and L0 (ρ̂x ) can be solved
finding a feedback L0 (ρ̂x ) such that:
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P +
P A0 (ρ̂x ) − L0 (ρ̂x )C0 < 0
V̇ (t) = (t)
˙ T P + (t)T P (t)
˙ <0
(25)
this yields to:
h
V̇ (t) = (t)T ΛA0 (ρ̂x )T Λ−1 − ΛC0T L0 (ρ̂x )T P
i
+P Λ−1 A0 (ρ̂x )Λ − L0 (ρ̂x )C0 Λ (t) +
T
φ(x(t)) − φ(x̂(t)) Λ−1 P (t) +
(26)
(t)T P Λ−1 φ(x(t)) − φ(x̂(t))
+vx (t)T Λ−1 P (t) + (t)T P Λ−1 vx (t) < 0
Considering the structure of Λ (Equation (21)) and the
fact that A0 (ρ̂x ) is triangular and C0 = [ 1 .. 0 ], a simple
calculation yields to:
ΛA0 (ρ̂x )T Λ−1 − ΛC0T L0 (ρ̂x )T =
h
iT
λ A0 (ρ̂x ) − L0 (ρ̂x )C0
(28)
(27)
Replacing this result in (26) and taking into account that
φ(x(t)) satisfies ||φ(x(t)) − φ(x̂(t))|| < δ||x(t) − x̂(t)||, the
(31)
and a parameter λ large enough, such that (28) is accomplished, similarly as in the classical high gain observation
strategy presented before in Theorem 1.
To compute L0 (ρ̂x ) as (20), Li are deduced from the dual
solution of the quadratic stability of an uncertain plant
developed in Olalla et al. [2009] and Feron et al. [1992],
where a framework for robust linear quadratic regulators
(LQRs) control for a convex model of power converters,
taking into account uncertainty in the parameters is presented.
In this work, the LQR control problem with uncertain
parameters is solved by using an LMI. The dual representation of such controller is used. Thus, the solution to
solve the state feedback K for the LQR problem is stated
in the following theorem:
Theorem 2. Consider the system:
ẋ(t) =Ax(t) + Bu(t)
y(t) =Cx(t)
(32)
where A ∈ Rn×n and B ∈ Rn×p contain uncertainties,
u(t) ∈ Rp is the system input and C ∈ Rq×n where q is
the number of the system outputs. Given the symmetric
matrix P > 0 ∈ Rn×n , the matrices Y and X and the
parameter matrices W > 0 and V = V T > 0, the optimal
feedback gain K that guarantees that the system (32)
is quadratically stable can be found by minimizing the
following expression:
min(trace(X) + trace(V P ))
subject to the following linear matrix inequalities:
AP + P AT + BY + Y T B T + I < 0,
X
W 1/2 Y
<0
Y T W 1/2
P
(33)
(34)
Once this minimization under constraints is solved, the
controller can be recovered by K = Y P −1 .
Remark 2. The polytopic application this theorem consists on replacing the constraints involving matrices A and
B by M constraints corresponding to the vertexes of the
polytope formed by Ai and Bi , with matrices Xi and Yi .
To obtain the observer gain L0 (ρ̂x ), the dual solution of the
LQR problem in Theorem 2 for the polytopic case (remark
2) is used to compute Li , which yields to the following
proposition:
Proposition 2. Consider the observer (18). Given the symmetric matrix P > 0 ∈ Rn×n , the matrices Yi and Xi and
the parameter matrices W and V = V T > 0, such that
the following expression is minimized:
min(trace(Xi ) + trace(V P ))
(35)
subject to the linear matrices inequalities:
ATi P + P Ai + C0T Yi + YiT C0 + I < 0
Xi
W 1/2 Yi
YiT W 1/2
P
ρ̂x = −
for i = 1, 2, the matrices Li = Yi P −1 guarantee that
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P + P A0 (ρ̂x ) − L0 (ρ̂x )C0 < 0
for all ρx ∈ [ρ, ρ].
This theorem satisfies Inequality (31). The parameters W
and V are chosen as:
10
W = 0.05, V =
(38)
01
To complete the observer design, the parameters in the
matrix Λ have to be chosen large enough to guarantee the
estimation error to converge to 0. In this thesis, λ = 150,
thus:
150 0
Λ=
(39)
0 1502
thus, L0 (ρ̂x ) and Λ have been designed such that Proposition 1 is satisfied.
(40)
φ(x̂(t)) =
(41)

r
dV (t)
r
−
+
1
+
ωA
(t)k
T
x̂
(t)
w
0 w
1


V (t)
cv V (t)

 cv

 r
dV (t)
1
−
−
ωAw (t)(k1 x̂2 (t) − k0 Tw ) x̂2 (t)
cv
V (t)
V (t)

However, since y(t) = x1 (t) is a measured signal, ρ̂x may
indeed be replaced by:
ρ̂x = −
1
ωAw (t)k1 ŷ(t)
V (t)
(42)
Remark 4. If the perturbation vx = 0, Inequality (28)
yields:
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P + P A0 (ρ̂x )
i
−L0 (ρ̂x ) C0 ) (t) + 2δ(t)T P (t) < 0
(43)
λ(t)T
h
As it has been shown in Proposition 2, it is possible to find
a feedback such Inequality (31) is accomplish, moreover,
the dual solution of Theorem 2 guarantees the quadratic
stability of an observer using such a feedback, thus the
first term on Inequality (43) satisfies:
(36)
(37)
1
ωAw (t)k1 x̂1 (t)
V (t)
and
λ(t)T
<0
Remark 3. Notice that observing (10)-(13) implies the
estimates:
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P + P A0 (ρ̂x )
i
−L0 (ρ̂x ) C0 ) (t) < −λα||(t)||2
(44)
h
for some constant values α > 0. Using this result and
bounding the second term, Inequality (44) yields to:
T
A0 (ρ̂x ) − L0 (ρ̂x )C0 P + P A0 (ρ̂x )
i
−L0 (ρ̂x ) C0 ) (t) + 2δ(t)T P (t)
λ(t)T
h
< −λα||(t)||2 + β||(t)||2
(45)
for some constant β > 0. Thus a bound value for λ can be
β
that ensures the exponential stability
obtained as λ > α
of (t).
Remark 5. A restriction of this synthesis is the fact that
the LQR dual adaptation to obtain the gains Li has
brought two more calibration parameters besides the values of Λ. Such parameters are the matrices V and W ,
present in the synthesis of the dual solution of Theorem 2,
presented as the Proposition 2.
5.1 Simulation results
Using the observed states x̂1 and x̂2 , the enclosed mass in
the cylinder is computed using the ideal gas law (6):
m̂ =
x̂1 V
rx̂2
(46)
The results are compared to the virtual engine model
proposed by [Rivas et al., 2012]. Figures 3 and 4 show the
cylinder pressure and temperature estimations during the
compression stroke when the valves are closed and Figure 5
shows the result of the mass estimation, the operating
conditions are N = 4000 rpm and IM EP = 8 bar. The
second simulation case is shown in Figures 6, 7 and 8,
where the operating conditions are changed to N = 1200
rpm and IM EP = 2 bar.
Fig. 5. Cylinder mass estimation. Case 1: N=4000 rpm
IMEP=8 bar, IT=2.25 CAD
Fig. 3. Cylinder pressure. Case 1: N=4000 rpm IMEP=8
bar, IT=2.25 CAD
Fig. 6. Cylinder pressure. Case 2: N=1200 rpm IMEP= 2
bar, IT=15 CAD
Fig. 4. Cylinder Temperature. Case 1: N=4000 rpm
IMEP=8 bar, IT=2.25 CAD
The vertical bars in the figures represent the interval where
the observer is valid: the observer is enabled only during
the compression (valves closed), in the remaining parts of
the cycle the observer is disabled and reset. One engine
cycle is enough to accurately observe the variables of
interest x̂1 and x̂2 .
The compression stroke might be short in comparison with
the whole engine cycle, limiting the available time for the
estimation. In this thesis, the interest is to compute the
enclosed mass before the ignition. The observer settling
time to compute the enclosed mass is 1.7 ms. At 1200 rpm
the compression stroke lasts 6.5 ms for an IT = 15 CAD,
at 5500 rpm the compression stroke lasts 1.4 ms for an
Fig. 7. Cylinder Temperature. Case 2: N=1200 rpm
IMEP= 2 bar, IT=15 CAD
IT = 13 CAD and at 4500 rpm, the compression stroke
lasts around 2.1 ms. It shows that even if the initial
error is important when the observer is initialized, the
estimated mass converges soon enough before the ignition
timing for an engine speed N < 4500 rpm, always that
the compression stroke lasts more than 1.7 ms. For engine
speeds above 4500 rpm, the convergence of the observer
before the ignition timing is not ensured. An example of
such limitation is shown in Figure 9, where a test at an
Fig. 8. Cylinder mass estimation. Case 2: N=1200 rpm
IMEP= 2 bar, IT=15 CAD
engine speed of 5500 rpm is presented. Only the enclosed
mass estimation is plotted. As the engine speed is high, the
duration of the compression is short: 1.4 ms. Thus, the
observer has not converged when the combustion starts,
and the mass estimation cannot be achieved.
Remark 6. Notice that if the IT is too advanced and/or
the IVC is too delayed, it might shorten the compression
stroke by some crank angle degrees.
Fig. 9. Cylinder mass estimation. Case 3: N=5500 rpm
IMEP= 12 bar, IT=13.875
6. CONCLUSION
This work presents a new method to estimate the cylinder
enclosed mass during the compression stroke when the
engine valves are closed. A high gain nonlinear observer
based on a parameter varying model of the cylinder temperature during the compression stroke is built and the
enclosed mass is computed using the observed temperature
and the cylinder pressure measurement through the ideal
gases law. The preliminary results have shown to be effective to handle the strong non linearity of the compression
model. One engine cycle computation is enough to obtain
the mass estimation.
7. NOMENCLATURE
All variables are in S.I Metric Units.
ω: Engine speed (rad/s)
IT : Ignition timing
Aw : Heat transfer wall area
CAD: Crank angle degrees
cv : Specific heat at constant volume
hc : Heat transfer coefficient for wall losses
Hp : Piston height
k0 : Calibration constant
k1 : Calibration constant
N : Engine speed (rpm)
m: Total mass in the combustion chamber
p: Pressure
r: Specific gases constant
T : Temperature
Tw : Wall temperature
U : Energy
V : Cylinder volume
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